# Is there a Euclidean algorithm in $k[x,x^{-1},y]$?

Let $$k$$ be a field of characteristic zero, and let $$k[x,x^{-1},y]$$ be the polynomial ring in $$x,x^{-1},y$$.

Is there a Euclidean algorithm in $$k[x,x^{-1},y]$$?

Two relevant questions are: this and this.

Any hints and comments are welcome!

## 1 Answer

A Euclidean domain (i.e. an integral domain with a Euclidean function that allows the Euclidean algorithm to find gcd's, like the absolute value for $$\Bbb Z$$, or the degree for $$k[x]$$) must necessarily be a principle ideal domain. The ideal $$(x+1, y)\subseteq k[x, x^{-1}, y]$$ is not principal. So there is no Euclidean algorithm on your ring.

• Thank you. Same answer for $k[x,y]$? – user237522 Jun 19 at 10:48
• @user237522 Exactly (except there one can use the ideal $(x, y)$, which looks a bit nicer). Imagine using the extended Euclidean algorithm to write $1 = f\cdot x + g\cdot y$. That's clearly not going to work. Your case requires a little more care since $x$ is invertible, but not much. – Arthur Jun 19 at 10:50
• Thank you. $k[x,x^{-1},y]$ is a UFD, isn't it? ($k[x,y]$ is a UFD en.wikipedia.org/wiki/Unique_factorization_domain). – user237522 Jun 19 at 11:55
• @user237522 Yes, I believe it is. Which in particular means that gcd's exist. You just can't use the Euclidean algorithm to get to them. – Arthur Jun 19 at 11:57
• Thank you. (I have also seen this argument). – user237522 Jun 19 at 12:04